{"id":1737,"date":"2017-07-10T08:00:36","date_gmt":"2017-07-10T12:00:36","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=1737"},"modified":"2017-07-10T07:47:04","modified_gmt":"2017-07-10T11:47:04","slug":"help-wanted-mathematics-tutor","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2017\/07\/10\/help-wanted-mathematics-tutor\/","title":{"rendered":"Help Wanted: Mathematics Tutor"},"content":{"rendered":"

By Priscilla Bremser, Contributing Editor<\/a>, Middlebury College<\/em><\/p>\n

\u201cCan you recommend a good math tutor?\u201d I hear this question from friends with children in local schools, academic support staff at my institution, and my own students. \u00a0Once or twice I\u2019ve even heard it from a student on the first day of class.\u00a0 Although tutoring has much in common with other educational settings, it presents its own opportunities and challenges.\u00a0 In this post, I explore why one-on-one instruction is so appealing as a supplement to classroom instruction, and how effective tutors make the most of tutoring sessions.<\/p>\n

As Lepper and Woolverton point out in setting the stage for “The Wisdom of Practice: Lessons Learned from the Study of Highly Effective Tutors” [2, p. 138], \u201ctutorials provide a venue for learning that is inherently more individualized, more immediate, and more interactive than most common school settings.\u201d \u00a0Specifically, individualization ensures more focused attention from both tutor and tutee. Immediacy allows for instantaneous feedback.\u00a0 Interactivity means that the tutor can make real-time decisions and adjustments as the student\u2019s comprehension level and emotional state become more clear.<\/p>\n

The authors go on to identify specific practices of expert tutors. \u00a0While the overview is limited to studies of tutors for elementary school students studying mathematics, many of the effective practices it describes are also applicable to secondary and college mathematics settings. \u00a0For example, \u201cour best tutors seem to prefer a Socratic to a more didactic approach\u201d [2, p. 146]. \u00a0Naturally this approach involves asking questions and providing hints rather than providing quick answers. \u00a0It also includes making a distinction between \u201cproductive\u201d and \u201cnonproductive\u201d errors [p. 147] and responding accordingly. \u00a0A productive error is one that the student can self-correct, with the long-term learning benefits<\/a> that ensue, while a nonproductive error is best corrected immediately by the tutor.<\/p>\n

Readers of How People Learn<\/i><\/a> and related works will recognize a metacognition theme in this observation of Lepper and Woolverton: \u00a0\u201cmore effective tutors are more likely to ask students to articulate what they are learning, to explain their reasoning and their answers, and to generalize or relate their work in the tutoring session to other contexts and problems.\u201d \u00a0An expert tutor, then, guides the interaction not only for strong communication with the tutee, but also to strengthen and reinforce learning.<\/p>\n

The question of tutor-student communication is a complex one. \u00a0In a research review, Graesser et al. [1, p.418] point out five \u201cillusions\u201d that tutors may hold. \u00a0These are the illusions of grounding, feedback accuracy, discourse alignment, student mastery, <\/i>and knowledge transfer.<\/i> They categorize the misunderstandings that tutors often have about their students\u2019 thinking. \u00a0Have you ever been asked whether you understood something, and said \u201cyes\u201d even though you weren\u2019t sure? You were giving inaccurate feedback, and your questioner may not have caught on. \u00a0Of course even a sincere \u201cyes, I understand\u201d may be inaccurate, as \u201cit is the knowledgeable students who tend to say \u2018No, I don\u2019t understand.\u2019\u00a0 This result suggests that deeper learners have higher standards of comprehension\u201d (p. 414).<\/p>\n

For an example of poor discourse alignment, note that \u201ctutors sometimes give hints, but the students do not realize they are hints\u201d [p. 418]. Now that is a reality check. \u00a0In our previous post<\/a>, Jess Ellis Hagman wrote, \u201cMathematics education research is the systematic study of the teaching and learning of mathematics.\u201d\u00a0 Sometimes a seemingly small detail emerging from such study can have profound implications.<\/p>\n

More from Graessner et al.: \u201cA good tutor is sufficiently skeptical of the student\u2019s level of understanding. \u2026 A good tutor assumes that the student understands very little of what the tutor says and that knowledge transfer approaches zero \u2026 (E)xpert tutors are more likely to verify that the student understands what the tutor expresses by asking follow-up questions or giving follow-up troubleshooting problems\u201d [1, p. 419].\u00a0 I recall working with an algebra student who insisted that he understood the relationship between the graphs of $y = x^2$ and $y = x^2 +2$, even though his graphs intersected. \u00a0Rather than pointing at the intersection and explaining my concern, I should have suggested that he add the graph of $y = x^2 + 1$ and tell me what he noticed.<\/p>\n

Given recent research on the effects of students\u2019 emotions and mindsets<\/a> on learning, how do good tutors attend to those factors? For one thing, while they are supportive and kind, they are sparing with praise. When these tutors do offer compliments, they refer to the work, not the person.\u00a0 The compliment might be an indirect one, such as a simple, \u201cThat was a hard problem you just did.\u201d \u00a0Good tutors also find ways to turn control over to their students by, for example, letting the tutee choose between two equally challenging problems [2].<\/p>\n

Many of the above observations about effective tutoring, and potential pitfalls, are relevant to considerations of classroom instruction, especially active learning<\/a> environments in which instructors have frequent, though short, interactions with individual students and small groups.\u00a0 In addition, faculty office hours are often sequences of tutoring sessions.\u00a0 Occasionally I\u2019ve had the sense that a meeting with a student didn\u2019t go well because I said too much or corrected an interesting mistake too soon. \u00a0The research seems to confirm my impressions.<\/p>\n

Still, tutoring is different from classroom instruction in significant ways. \u00a0Most obviously, perhaps, tutoring usually happens when someone determines that special intervention is required.\u00a0 A student is struggling, or not doing as well as expected. \u00a0Perhaps the student\u2019s parents see tutoring as a way to improve grades or test scores for college applications. Under these conditions, it is especially important for the tutor to attend to the student\u2019s affective state.<\/p>\n

Additionally, although the appeal of tutoring as a remedy springs from the one-on-one nature of tutoring sessions, there are usually other people on the periphery. There\u2019s the classroom instructor, who may have recommended tutoring, or may not know that it is happening. \u00a0Perhaps the student\u2019s parents are involved. Many school districts coordinate tutoring programs<\/a> in cooperation with local organizations.\u00a0 It seems reasonable to conclude that communication challenges come along with those added relationships.<\/p>\n

For one thing, the tutor may not know or understand the instructor\u2019s learning objectives for the student. A peer tutor for my Calculus I students may have taken AP Calculus in high school, which can be a very different course from mine. \u00a0A volunteer tutor in a public school might remember shortcuts for working with fractions, while the teacher wants to Nix the Tricks<\/a>.<\/p>\n

Further, the tutor might not have a deep understanding of the relevant mathematical content. \u00a0As a sophomore in college, I signed up to be a peer tutor. \u00a0A junior came to me for help with multivariable calculus.\u00a0 She was baffled by parametric curves, which hadn\u2019t been covered in my multivariable course the previous year. \u00a0At the time I was mortified, feeling somehow that I\u2019d failed personally. \u00a0But she and I tried to work through that section of the textbook together, which (I now recognize) was probably good for both of us. \u00a0According to [1], \u201ctutors in …same-age and cross-age collaborations tend to learn more than the tutees\u201d (p. 412). \u00a0It\u2019s probably important that I knew what I didn\u2019t know about parametric curves. In contrast, a colleague once overheard a peer tutor say, \u201cthe individual terms of the series go to zero, so it has to converge\u201d in our department common room. \u00a0Fortunately, our peer tutors now undergo appropriate training before they start.<\/p>\n

Can I recommend a good math tutor?\u00a0 Yes, but I would want that tutor to get training first.\u00a0 It wouldn’t hurt to also read [1] and [2]!\u00a0 (Other resource suggestions are welcome in the comments.)\u00a0 Good tutors know that showing and telling should be used sparingly, and only after careful listening.<\/p>\n

Thanks to Steve Klee for directing me<\/em>\u00a0to\u00a0<\/em>[2].<\/p>\n

REFERENCES<\/strong><\/p>\n

[1] Graesser, A. C., D\u2019Mello, S., & Cade, W. (2011). Instruction based on tutoring. Handbook of research on learning and instruction<\/i>, 408-426.<\/p>\n

[2] Lepper, M. R., & Woolverton, M. (2002). The wisdom of practice: Lessons learned from the study of highly effective tutors. Improving academic achievement: Impact of psychological factors on education<\/i>, 135-158.<\/p>\n

 <\/p>\n

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By Priscilla Bremser, Contributing Editor, Middlebury College \u201cCan you recommend a good math tutor?\u201d I hear this question from friends with children in local schools, academic support staff at my institution, and my own students. \u00a0Once or twice I\u2019ve even … Continue reading →<\/span><\/a><\/p>\n

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